Life In 19x19

Asked differently for an example, if we have ^2* + v + v + * + X, can X be on the board so that we are required to ignore ^2* + v + v + * = 0 and start playing in X because any other move, in ^2* + v + v + *, would be a mistake?

RobertJasiek wrote:Asked differently for an example, if we have ^2* + v + v + * + X, can X be on the board so that we are required to ignore ^2* + v + v + * = 0 and start playing in X because any other move, in ^2* + v + v + *, would be a mistake?

We are not required to ignore it, but we can ignore it. It makes our job easier. A move in ^2* + v + v + * might be a mistake. Why take the chance?

BTW, there are a number of problems where there is a miai and playing in the miai would be a mistake. Such positions occur in actual play, as well.

Seems I need to find an example myself...

Click Here To Show Diagram Code

[go]$$W Miai plus miny, White to play in a corridor
$$ -------------------------------
$$ . X W . . . O B . . X . . B B O .
$$ . X X X X X O O O O X O O O O O .
$$ . . . X W . O B . . X . . . . . .
$$ . . . X X X O O O O X . X . . . .
$$ . . . . . . . . . . . . . . . . .[/go]

Unmarked stones are invulnerable.

Bill Spight wrote:

$$W Miai plus miny, White to play in a corridor
$$ -------------------------------
$$ . X W . . . O B . . X . . B B O .
$$ . X X X X X O O O O X O O O O O .
$$ . . . X W . O B . . X . . . . . .
$$ . . . X X X O O O O X . X . . . .
$$ . . . . . . . . . . . . . . . . .

Click Here To Show Diagram Code

[go]$$W Miai plus miny, White to play in a corridor
$$ -------------------------------
$$ . X W . . . O B . . X . . B B O .
$$ . X X X X X O O O O X O O O O O .
$$ . . . X W . O B . . X . . . . . .
$$ . . . X X X O O O O X . X . . . .
$$ . . . . . . . . . . . . . . . . .[/go]

Unmarked stones are invulnerable.

Very nice example for the requirement to ignore 0 sums! I have tried a fraction, *, up or down but your miny does it! The unfortunate consequence is that infinitesimals must not be avoided even for seemingly simple considerations related to equal options.

Click Here To Show Diagram Code

[go]$$W Correct, smaller count = -5
$$ -------------------------------
$$ . X W . . . O B . . X 1 . B B O .
$$ . X X X X X O O O O X O O O O O .
$$ . . . X W . O B . . X . . . . . .
$$ . . . X X X O O O O X . X . . . .
$$ . . . . . . . . . . . . . . . . .[/go]

The remaining local endgames sum to 0. It is correct for White to ignore them, starting elsewhere in the miny.

Click Here To Show Diagram Code

[go]$$W Mistake, larger count = -4
$$ -------------------------------
$$ . X W . . 1 O B . . X 2 3 B B O .
$$ . X X X X X O O O O X O O O O O .
$$ . . . X W . O B . 4 X . . . . . .
$$ . . . X X X O O O O X . X . . . .
$$ . . . . . . . . . . . . . . . . .[/go]

The remaining local endgames sum to 0. It is wrong for White to attack the local endgames whose initial sum was 0.

Click Here To Show Diagram Code

[go]$$W Miai plus down, White to play in a corridor
$$ -------------------------------
$$ . X W . O O O . . X . . B O . .
$$ . X X X X X O . . X O O O O O .
$$ . . . X W . O . . X . . . . . .
$$ . . . X X X O . . X . X . . . .
$$ . . . . . . . . . . . . . . . . .[/go]

Here is an example with White's only correct play in the down.

QUESTIONS 12:

What is an example for attacking MINY-x|0^n being better than attacking MINY-x|0^(n+1)?

Click Here To Show Diagram Code

[go]$$B MINY-2|0 and MINY-2|0^2
$$ . . . . . . . . .
$$ . O O O O O O . .
$$ X . . . X X O . .
$$ . O O O O O O . .
$$ . O O O O O O O .
$$ X . . . . X X O .
$$ . O O O O O O O .
$$ . . . . . . . . .[/go]

What is an example for attacking MINY-x|0^n being better than attacking a DOWN-d or DOWN-d-STAR corridor?

Click Here To Show Diagram Code

[go]$$B MINY-2|0 and DOWN
$$ . . . . . . . . .
$$ . O O O O O O . .
$$ X . . . X X O . .
$$ . O O O O O O . .
$$ . O O O O . . . .
$$ X . . X O . . . .
$$ . O O O O . . . .
$$ . . . . . . . . .[/go]

EDIT

White 1 produces a position where Black needs to attack the shorter corridor. Attacking the longer corridor lets White win.

Again, after White 1 Black must not attack the down.

Ah, the difference game, very nice!

QUESTIONS 13:

Mathematical Go Endgames creates the impression of positional judgement being possible by rounding and explains it for numbers, ups, downs, tinys, minys by comparing them to each other.

How to round if also 0^n|TINY-x or MINY-x|0^n are involved?

How do 0^n|TINY-x and TINY-x compare?

How do 0^n|TINY-x and TINY-y compare (x<>y)?

How do 0^n|TINY-x and 0^n|TINY-y compare (x<>y)?

How do 0^m|TINY-x and 0^n|TINY-x compare (m<>n)?

How do 0^m|TINY-x and 0^n|TINY-y compare (m<>n, x<>y)?

What is the white attacker's incentive in 0^n|TINY-x (n>1)?

What is the white attacker's incentive in 0|TINY-x?

Why is 0^n|TINY-x about UP-n + STAR^n? Slightly smaller or larger? What does this tell us for rounding?

How do uptimals compare to ordinary infinitesimals?

Without answering these questions, positional judgement by rounding is a myth, unless there are only numbers, ups/dows, stars, tinys/minys.

For the theory of atomic weights see On Number and Games and Winning Ways.

QUESTIONS 14:

These questions are about a group invading several corridors.

Has Mathematical Go Endgames only studied empty corridors?

Where to play as attacker / defender when one string with one socket invades blocked and / or unblocked empty corridors?

Where to play as attacker / defender when one group with multiple sockets invades blocked and / or unblocked empty corridors?

Information on this is encrypted in the proofs and I could not decipher it yet.